# A strong form of the Nagell-Lutz theorem

The motivation of this question can be found in

Is it possible to say that every point $P$ in $C(ℚ)$ other than the 'basis' is of finite order?

Given the elliptic curve: $$C:y²=x³+ax+b$$

for $a,b∈ℤ$.

Let $O$ be the identity element of $C(ℚ)$. I know about the Nagell-Lutz theorem [T. Nagell, Solution de quelque problèmes dans la théorie arithmétique des cubiques planes du premier genre, Wid. Akad. Skrifter Oslo I (1935), No. 1, 1-25, E. Lutz, Sur l'equation $y²=x³-Ax-B$ dans les corps $p$-adic, J. Reine Angew. Math. 177 (1937), 238-247.] in its explicit form. However, I found a strong form of the Nagell-Lutz theorem saying that if $T∈C(ℚ)^{tors}$, then $2T=O$. I do not understand how I can obtain the formula $2T=O$ from the standard Nagell-Lutz theorem.

• Where did you find this "strong form of the Nagell Lutz theorem"? The curve $y^2 = x^3 - 432x + 8208$ has a point $T=(-12,108)$ of order $5$. – Álvaro Lozano-Robledo Apr 8 '13 at 13:36
• @ÁlvaroLozano-Robledo: Please see ucl.ac.uk/~ucahmki/pofo.pdf – Germany Apr 8 '13 at 14:11
• I do not see anywhere a statement that says "If $T\in C(\mathbb{Q})^\text{tors}$, then $2T=\mathcal{O}$". Notice the OR ELSE $y^2| D$. – Álvaro Lozano-Robledo Apr 8 '13 at 14:50
• @ÁlvaroLozano-Robledo: Thank you very much for your nice answer in (math.stackexchange.com/questions/349922/…) . How I can express the point at infinity (the identity element of $C(ℚ)$) in that representation. – Germany Apr 9 '13 at 6:38
• $\mathcal{O}=0\cdot P_1+\cdots+0\cdot P_r + \mathcal{O}$. – Álvaro Lozano-Robledo Apr 9 '13 at 14:40